Moduli Spaces of Rank 2 Acm Bundles on Prime Fano Threefolds

MODULI SPACES OF RANK 2 ACM BUNDLES ON PRIME FANO THREEFOLDS

MARIA CHIARA BRAMBILLA AND DANIELE FAENZI

Abstract. Given a non-hyperelliptic prime Fano threefold X, we prove the existence of all rank 2 ACM vector bundles on X by deformation of semistable sheaves.

1. Introduction

A vector bundle F on a smooth polarized variety (X,HX ) has no in- k termediate cohomology if: H (X,F ⊗ OX (t HX )) = 0 for all t ∈ Z and 0 < k < dim(X). These bundles are also called arithmetically Cohen- Macaulay (ACM), because they correspond to maximal Cohen-Macaulay modules over the associated graded ring. It is known that an ACM bundle n must be a direct sum of line bundles if X = P (see [Hor64]), or a direct sum of line bundles and (twisted) spinor bundles if X is a smooth quadric n hypersurface in P ([Knö87],[Ott89]). On the other hand, there exists a complete classification of varieties admitting, up to twist, a finite number of isomorphism classes of indecomposable ACM bundle see [EH88], [BGS87]. Only five cases exist besides rational normal curves, projective spaces and quadrics. For varieties which are not in this list, the problem of classifying ACM bundles has been taken up only in some special cases. For instance, on general hypersurfaces of dimension at least 3, a full classification of ACM bundles of rank 2 is available, see [MKRR07a], [MKRR07b], [CM04], [CM00], [Mad00]. For dimension 2 and rank 2, a partial classification can be found in [Fae08], [CF06]. The case of smooth Fano threefolds X with Picard number 1 has also ∼ been extensively studied. In this case one has Pic(X) = hHX i, with HX ample, and KX = −iX HX , where the index iX satisfies 1 ≤ iX ≤ 4. Recall ∼ 3 that iX = 4 implies X = P and iX = 3 implies that X is isomorphic to a smooth quadric. Thus, the class of ACM bundles is completely understood in these two cases. In contrast to this, the cases iX = 2, 1 are highly nontrivial. First of all, there are several deformation classes of these varieties, see [Isk77], [Isk78], [IP99]. A different approach to the classification of these varieties was pro- posed by Mukai, see for instance [Muk88], [Muk89], [Muk95].

2000 Mathematics Subject Classiﬁcation. Primary 14J60. Secondary 14F05, 14D20, 13C14. Key words and phrases. Prime Fano threefolds. Moduli space of vector bundles. Bun- dles with no intermediate cohomology. Arithmetically Cohen-Macaulay sheaves. Both authors were partially supported by Italian MIUR funds. 1 2 MARIA CHIARA BRAMBILLA AND DANIELE FAENZI

In second place, it is still unclear how to characterize the invariants of ACM bundles: in fact the investigation has been deeply carried out only in the case of rank 2. For iX = 2, the classification was completed in [AC00]. For iX = 1, a result of Madonna (see [Mad02]) implies that if a rank 2 ACM bundle F is defined on X, then its second Chern class c2 must take values in {1, . . . , g + 3} if c1(F ) = −1, or in {2, 4} if c1(F ) = 0. Partial existence results are given in [Mad00], [AF06]. In third place, the set of ACM rank 2 bundles can have positive dimension. A natural point of view is to study them in terms of the moduli space MX (2, c1, c2) of (Gieseker)-semistable rank 2 sheaves F with c1(F ) = c1, c2(F ) = c2, c3(F ) = 0. For iX = 2, such moduli space has been mostly studied when X is a smooth cubic threefold, see for instance [MT01], [IM00a], [Dru00], see also [Bea02] for a survey. If the index iX equals 1, the threefold X is said to be prime, and one 3 defines the genus of X as g = 1 + HX /2. In this case, some of the rele- vant moduli spaces MX (2, 1, c2) are studied in [IM00b] (for g = 3), [IM04], [IM07c], [BF07] (for g = 7), [IM07a], [IM00a] (for g = 8), [IR05] (for g = 9), [AF06] (for g = 12). The goal of our paper is to provide the classification of rank 2 ACM bundles F on a smooth prime Fano threefold X, i.e. in the case iX = 1. Note that we can assume c1(F ) ∈ {0, 1}. Combining our existence theorems (namely Theorem 3.6 and Theorem 4.10) with the results of Madonna and others mentioned above, we obtain the following classification.

Theorem. Let X be a smooth prime Fano threefold, with −KX very ample. Then there exists an ACM vector bundle F of rank 2

g+2 i) with c1(F ) = 1 if and only if c2(F ) = 1 or d 2 e ≤ c2(F ) ≤ g + 3, ii) with c1(F ) = 0 if and only if c2(F ) = 2, 4, where in case (i) existence holds under the further assumption that X contains a line L with normal bundle OL ⊕ OL(−1).

Note that the assumption that −KX is very ample (the threefold X is thus called non-hyperelliptic) excludes two families of prime Fano threefolds, one with g = 2, the other with g = 3. These two cases will be discussed in a forthcoming paper. On the other hand, the assumption that X contains a line L with normal bundle OL ⊕ OL(−1) (in this case we will say that X is ordinary) is always veriﬁed if g ≥ 9 unless X is the Mukai-Umemura threefold of genus 12. However this condition is veriﬁed by a general prime Fano threefold of any genus, see section 2.3 for more details. The paper is organized as follows. In the following section we give some preliminary notions. Section 3 is devoted to the proof of part (i) of the main theorem. Section 4 concerns the case (ii) of the main theorem. We end the paper describing some properties of the moduli space MX (2, 0, 4).

Acknowledgements. We would like to thank Atanas Iliev and Giorgio Ottaviani for several useful remarks. ACM BUNDLES ON PRIME FANO THREEFOLDS 3

2. Preliminaries Given a smooth complex projective n-dimensional polarized variety (X,HX ) and a sheaf F on X, we write F (t) for F ⊗ OX (tHX ). Given a subscheme Z of X, we write FZ for F ⊗ OZ and we denote by IZ,X the ideal sheaf of Z in X, and by NZ,X its normal sheaf. We will frequently drop the second subscript. k Given a pair of sheaves (F,E) on X, we will write extX (F,E) for ˇ k the dimension of the Cech cohomology group ExtX (F,E), and similarly hk(X,F ) = dim Hk(X,F ). The Euler characteristic of (F,E) is defined as P k k χ(F,E) = k(−1) extX (F,E) and χ(F ) is defined as χ(OX ,F ). We denote by p(F, t) the Hilbert polynomial χ(F (t)) of the sheaf F . The degree n−1 deg(L) of a divisor class L is defined as the degree of L · HX .

2.1. ACM sheaves. Let (X,HX ) be a n-dimensional polarized variety, and m assume HX very ample, so we have X ⊂ P . We denote by IX the ideal of m X in P , and by R(X) the coordinate ring of X. Given a sheaf F on X, we deﬁne the following R(X)-modules:

k M k H∗(X,F ) = H (X,F (t)), for each k = 0, . . . , n. t∈Z The variety X is said to be arithmetically Cohen-Macaulay (ACM) if 1 m m R(X) is a Cohen-Macaulay ring. This is equivalent to H∗(P , IX,P ) = 0 k m and H∗(P , OX ) = 0 for 0 < k < n. A sheaf F on a variety X is called locally Cohen-Macaulay if for any point x ∈ X we have depth(Fx) = dim(Ox). Deﬁnition 2.1. A sheaf F on a ACM variety X is called ACM (arithmetically Cohen-Macaulay) if F is locally Cohen-Macaulay and it has no intermediate cohomology, i.e.: k (2.1) H∗(X,F ) = 0 for any 0 < k < 0 . By [CH04, Proposition 2.1], there is a one-to-one correspondence between ACM sheaves on X and graded maximal Cohen-Macaulay modules on R(X), 0 given by F 7→ H∗(X,F ). If X is smooth and F is locally free, then F is ACM if and only if the cohomological condition (2.1) holds. Moreover if X is smooth, any ACM sheaf is locally free.

2.2. Stability and moduli spaces. Let us now recall a few well-known facts about semistable sheaves on projective varieties. We refer to the book [HL97] for a more detailed account of these notions. Let (X,HX ) be a smooth complex projective n-dimensional polarized variety. We recall that a torsionfree coherent sheaf F on X is (Gieseker) semistable if for any coherent subsheaf E, with 0 < rk(E) < rk(F ), one has p(E, t)/ rk(E) ≤ p(F, t)/ rk(F ) for t 0. The sheaf F is called stable if the inequality above is always strict. The slope of a sheaf F of positive rank is defined as µ(F ) = deg(c1(F ))/ rk(F ), where c1(F ) is the first Chern class of F . We say that F is normalized if −1 < µ(F ) ≤ 0. We recall that a torsionfree coherent sheaf F is µ-semistable if for any coherent subsheaf E, with 0 < rk(E) < rk(F ), 4 MARIA CHIARA BRAMBILLA AND DANIELE FAENZI one has µ(E) < µ(F ). The sheaf F is called µ-stable if the above inequality is always strict. The two notions are related by the following implications: µ-stable ⇒ stable ⇒ semistable ⇒ µ-semistable Recall that by Maruyama’s theorem, see [Mar80], if dim(X) = n ≥ 2 and F is a µ-semistable sheaf of rank r < n, then its restriction to a general hypersurface of X is still µ-semistable. We will make use of different notions of moduli spaces. Let us introduce here some notation. We denote by MX (r, c1, . . . , cn) the moduli space of S-equivalence classes of rank r torsionfree semistable sheaves on X with k,k Chern classes c1, . . . , cn, where ck lies in H (X). The Chern class ck will be denoted by an integer as soon as Hk,k(X) has dimension 1. We will drop the last values of the classes ck when they are zero. µ We denote by MX (r, c1, . . . , cr) the moduli space of S-equivalence classes of µ-semistable sheaves with fixed rank and Chern classes. Given a filtration of a sheaf F , we denote by gr(F ) the associated graded object. The moduli space of simple sheaves with fixed rank and Chern classes will be denoted by SplX (r, c1, . . . , cn). For an account on this moduli space we refer to [AK80], [KO89], [LO87], see also [Muk84]. g We denote by Hd (X) the union of components of the Hilbert scheme of closed Z subschemes of X with Hilbert polynomial p(OZ , t) = dt + 1 − g, containing integral curves of degree d and arithmetical genus g.

2.3. Prime Fano threefolds. Let now X be a smooth projective variety of dimension 3. Recall that X is called Fano if its anticanonical divisor class −KX is ample. A Fano threefold is called non-hyperelliptic if −KX is very ample. We say that X is prime if the Picard group is generated by the canonical ∼ ∼ divisor class KX . If X is a prime Fano threefold we have Pic(X) = Z = hHX i, where HX = −KX is ample. These varieties are classified up to deformation, see for instance [IP99, Chapter IV]. The number of deformation classes is 10. It is well-known that a smooth prime Fano threefold is ACM. One defines the genus of a prime Fano threefold X as the integer g such 3 that deg(X) = HX = 2 g − 2. It is known that the genus of a non- hyperelliptic prime Fano threefolds takes values in {3,..., 10, 12}, while there are two families (one for g = 2, another for g = 3) that consist of hyperelliptic threefolds. A hyperelliptic prime Fano threefold of genus 3 is a 4 flat specialization of a quartic hypersurface in P , see [Man93] and references therein. It is well known that any prime Fano threefold X contains lines and conics. 0 The Hilbert scheme H1 (X) of lines contained in X is a projective curve. It is well known that the surface swept out by the lines of a prime Fano threefold X is linearly equivalent to the divisor rHX , for some r ≥ 2, see e.g. [IP99, Table at page 76]. Moreover every line meets finitely many other lines, see [Isk78, Theorem 3.4, iii] and [Man93]. 0 A prime Fano threefold X is said to be exotic if the Hilbert scheme H1 (X) has a nonreduced component. By [Isk78, Lemma 3.2], this is equivalent to the fact that for any line L ⊂ X of this component, the normal bundle NL splits as OL(1)⊕OL(−2). It is well-known that a general prime anticanonical ACM BUNDLES ON PRIME FANO THREEFOLDS 5 threefold X is not exotic. On the other hand, for g ≥ 9, the results of [GLN06] and [Pro90] imply that X is nonexotic unless g = 12 and X is the Mukai-Umemura threefold, see [MU83]. We say that a prime Fano threefold 0 X is ordinary if the Hilbert scheme H1 (X) has a reduced component. This is equivalent to the fact that there exists a line L ⊂ X whose normal bundle NL splits as OL ⊕ OL(−1). If X is a non-hyperelliptic prime Fano threefold, the Hilbert scheme 0 H2 (X) of conics contained in X is a projective surface, and a general conic C in X has trivial normal bundle, see [Isk78, Proposition 4.3 and Theorem 4.4]. Moreover, the threefold X is covered by conics. Moreover if X is a 0 general prime Fano threefold, then H2 (X) is a smooth surface, see [IM07b] for a survey. Recall also that a smooth projective surface S is a K3 surface if it has trivial canonical bundle and irregularity zero. A general hyperplane section S of a prime Fano threefold X is a K3 surface, polarized by the restriction HS of HX to S. If X has genus g, then S has (sectional) genus g, and degree 2 HS = 2 g − 2. If X is non-hyperelliptic, by Moishezon’s theorem [Mo˘ı67], ∼ we have Pic(S) = Z = hHSi. Note that a general hyperplane section of a hyperelliptic prime Fano threefold is still a K3 surface of Picard number 1 if g = 2. This is no longer true in the other hyperelliptic case, i.e. for g = 3. Indeed, let X be a double 4 cover of a smooth quadric in P ramified along a general octic surface. Then the general hyperplane section is a K3 surface of Picard number 2.

2.4. Summary of basic formulas. From now on, X will be a prime smooth Fano threefold of genus g, polarized by HX . Let F be a sheaf of (generic) rank r on X with Chern classes c1, c2, c3. Recall that these classes will be denoted by integers, since Hk,k(X) is generated by the class of HX (for k = 1), the class of a line contained in X (for k = 2), the class of a closed point of X (for k = 3). We will say that F is an r-bundle if it is a vector bundle (i.e. a locally free sheaf) of rank r. We recall that the discriminant of F is 2 (2.2) ∆(F ) = 2 r c2 − (r − 1)(2 g − 2) c1. Bogomolov’s inequality, see for instance [HL97, Theorem 3.4.1], states that if F is a µ-semistable sheaf, then we have: (2.3) ∆(F ) ≥ 0. Applying to the sheaf F the theorem of Hirzebruch-Riemann-Roch, we obtain the following formulas: 11 + g g − 1 1 g − 1 1 1 χ(F ) = r + c + c2 − c + c3 − c c + c , 6 1 2 1 2 2 3 1 2 1 2 2 3 1 χ(F,F ) = r2 − ∆(F ). 2 We recall by [Muk84], see also [HL97, Part II, Chapter 6] that, given a simple sheaf F of rank r on a K3 surface S of genus g, with Chern classes c1, c2, the dimension at the point [F ] of the moduli space SplS(r, c1, c2) is: (2.4) ∆(F ) − 2 (r2 − 1), 6 MARIA CHIARA BRAMBILLA AND DANIELE FAENZI where ∆(F ) is still deﬁned by (2.2). If the sheaf F is stable, this value also equals the dimension at the point [F ] of the moduli space MS(r, c1, c2). We recall now the well-known Hartshorne-Serre correspondence, relating vector bundles of rank 2 over X with subvarieties Z of X of codimension 2. For more details we refer to [Har74], [Har78], [Har80, Theorem 4.1], see also [Arr07] for a survey. Proposition 2.2. Let X be a smooth prime Fano threefold, and ﬁx the integers c1, c2. Then we have a one-to-one correspondence between (1) the pairs (F, s), where F is a rank 2 vector bundle on X with ci(F ) = ci and s is a global section of F whose zero locus has codimension 2, and (2) the locally complete intersection Cohen-Macaulay curves Z ⊂ X, ∼ with ωZ = OZ (c1 − 1), and of degree c2.

Recall that in the above correspondence Z has arithmetical genus pa(Z) = d(1−c1) 1 − 2 . The zero locus of a nonzero global section s of a rank 2 vector bundle F has codimension 2 if F is globally generated and s is general, or if H0(X,F (−1)) = 0. Lemma 2.3. Let X be a smooth prime Fano threefold and let s be a nonzero global section of a locally free sheaf F on X, whose zero locus is a curve D ⊂ X. Then we have: 1 ∼ 1 (2.5) H∗(X,F ) = H∗(X, ID(1)). In particular F is ACM if and only if D is projectively normal. Proof. The section s gives the exact sequence:

(2.6) 0 → OX → F → ID(c1(F )) → 0, and taking cohomology we obtain the required isomorphism (2.5). m ` Let now X ⊂ P and D ⊂ P with ` ≥ 2, m ≥ 4. By deﬁnition, the curve 1 ` D is projectively normal if and only if H ( , I ` ) = 0. Clearly, this is ∗ P D,P 1 m m equivalent to H∗(P , ID,P ) = 0. We have the exact sequence:

m m 0 → IX,P → IC,P → IC,X → 0. Taking cohomology, we get that D is projectively normal if and only if 1 H∗(X, ID(1)) = 0, which is equivalent to F being ACM by (2.5) and Serre duality. 2.5. ACM bundles of rank 2. In this section, we recall Madonna’s result in the case of bundles of rank 2 on a smooth prime Fano threefold. Proposition 2.4 ([Mad02]). Let X be a smooth prime Fano threefold of genus g, and let F be a normalized ACM 2-bundle on X. Then the Chern classes c1 and c2 of F satisfy the following restrictions:

c1 = 0 ⇒ c2 ∈ {2, 4},

c1 = −1 ⇒ c2 ∈ {1, . . . , g + 3}.

Remark 2.5. Let F , c1, c2 be as above, and t0 be the smallest integer t such that H0(X,F (t)) 6= 0. In [Mad02] the author computes the following values of t0: ACM BUNDLES ON PRIME FANO THREEFOLDS 7 a) if (c1, c2) = (−1, 1), then t0 = 0, b) if (c1, c2) = (0, 2), then t0 = 0, c) if (c1, c2) = (−1, c2), with 2 ≤ c2 ≤ g + 2, then t0 = 1, d) if (c1, c2) = (0, 4), then t0 = 1, e) if (c1, c2) = (−1, g + 3), then t0 = 2. We observe that F is not semistable in cases (a) and (b), and strictly µ-semistable in case (b). In fact, these two cases correspond respectively to lines and conics contained in X, and the existence of F is well-known. On the other hand, in the remaining cases, if F exists then it is a µ-stable sheaf. The following lemma (see [BF07, Lemma 3.1]) gives a small (sharp) restriction on the values in Madonna’s list. Lemma 2.6. Let X be a smooth non-hyperelliptic prime Fano threefold of genus g and set: g + 2 (2.7) m = . g 2

Then the moduli space MX (2, 1, d) is empty for d < mg. In particular, we get the further restriction c2 ≥ mg in case (c). Remark 2.7. The assumption non-hyperelliptic cannot be dropped. Indeed if X is a hyperelliptic Fano threefold of genus 3, then the moduli space 4 MX (2, 1, 2) is not empty. Indeed let Q ∈ P be a smooth quadric and π : X → Q be a double cover ramiﬁed along a general octic surface. Let S be the spinor bundle on Q and set F = π∗(S ). Then F is a stable vector bundle on X lying in MX (2, 1, 2). Notice that the restriction FS to any hyperplane section S ⊂ X is strictly semistable.

3. Bundles with odd first Chern class Throughout the paper, we denote by X a smooth non-hyperelliptic prime Fano threefold of genus g. In this section we will prove the existence of the semistable bundles appearing in the (restricted) Madonna’s list, whose ﬁrst Chern class is odd. We will study ﬁrst the case of minimal c2 and then we proceed recursively. We conclude this section studying the curves arising as the zero locus of a global section of a rank 2 ACM bundle.

3.1. Moduli of ACM 2-bundles with minimal c2. In this section we study the moduli space with odd c1 and minimal c2. Let F be a sheaf of ∼ ∗ rank 2. If the sheaf F is locally free, we have F = F (c1(F )). Notice that, if c1(F ) is odd, F is µ-stable as soon as it is µ-semistable. The following theorem is well known. Up to the authors’ knowledge, there is no proof of this result, other than a case-by-case analysis. We refer e.g. to [Mad00] for g = 3, [Mad02] for g = 4, 5, [Gus82] for g = 6, [IM04], [IM07c], [Kuz05], for g = 7, [Gus83], [Gus92], [Muk89] for g = 8, [IR05] for g = 9, [Muk89] for g = 10, [Kuz96] (see also [Sch01], [Fae07]) for g = 12. Theorem 3.1. Let X be a smooth non-hyperelliptic prime Fano threefold l g+2 m of genus g, and set mg = 2 . Then MX (2, 1, mg) is not empty. 8 MARIA CHIARA BRAMBILLA AND DANIELE FAENZI

The following result appears in [BF07, Proposition 3.4], and is essentially due to Iliev and Manivel. Proposition 3.2. Let X be a smooth non-hyperelliptic prime Fano threefold of genus g and set mg as in (2.7). Then any sheaf F in MX (2, 1, mg) is stable, locally free and it satisﬁes: k H (X,F (−1)) = 0, for all k ∈ Z, Hk(X,F ) = 0, for all k ≥ 1, 0 h (X,F ) = g − mg + 3.

Moreover i f g ≥ 6, then any sheaf F in MX (2, 1, mg) is globally generated and ACM. In the next lemmas we review some results concerning the moduli space MX (2, 1, mg). Though all the statements are essentially well-known, we present a proof in some cases where we outline a more direct argument, or when no explicit reference is available in the literature. Lemma 3.3. Let X be a smooth non-hyperelliptic prime Fano threefold of 0 genus g ≥ 6, and let F and F be sheaves in MX (2, 1, mg). Then we have 2 0 ExtX (F,F ) = 0. In particular the space MX (2, 1, mg) is smooth. If g is even, this implies that MX (2, 1, mg) consists of a single point. Proof. Recall by Proposition 3.2 that F and F 0 are stable locally free globally generated ACM sheaves. We can thus write the natural evaluation exact sequence: 0 0 0 (3.1) 0 → K → H (X,F ) ⊗ OX → F → 0, where the sheaf K is locally free, and we have:

rk(K) = g − mg + 1, c1(K) = −1. Note that K is a stable bundle by Hoppe’s criterion, see for instance [AO94, Theorem 1.2], [Hop84, Lemma 2.6]. Indeed, note that H0(X,K) = 0, and we have −1 < µ(∧pK) < 0. By the inclusion: ∧pK,→ ∧p−1K ⊗ H0(X,F ), we obtain recursively H0(X, ∧pK) = 0 for all p ≥ 0. Note that, by stability and by Proposition 3.2, we have Hk(X,F ∗) = 0 all k. Thus, tensoring (3.1) by F ∗, we obtain: 2 0 2 ∗ 0 3 ∗ 0 ∗ ∗ extX (F,F ) = h (X,F ⊗ F ) = h (X,F ⊗ K) = h (X,K ⊗ F ) = 0, ∗ ∗ where the last equality holds by stability, indeed c1(K ⊗ F ) = mg −g+1 < 0 for g ≥ 6. Note that, when g is even, we have χ(F,F 0) = 1. This gives 0 0 HomX (F,F ) 6= 0. But a nonzero morphism F → F has to be an isomorphism. This concludes the proof. Lemma 3.4. Let X be a smooth non-hyperelliptic prime Fano threefold of genus g ≥ 6 and g odd. Then the space MX (2, 1, mg) is ﬁne and isomorphic to a smooth irreducible curve. ACM BUNDLES ON PRIME FANO THREEFOLDS 9

Proof. Given a sheaf F in MX (2, 1, mg), by Lemma 3.2, it is an ACM globally generated vector bundle. By Lemma 3.3, the moduli space is smooth 1 and, since χ(F,F ) = 0, we have extX (F,F ) = homX (F,F ) = 1. Thus MX (2, 1, mg) is a nonsingular curve. Recall now that the obstruction to the existence of a universal sheaf on X × MX (2, 1, mg) corresponds to an element of the Brauer group of MX (2, 1, mg), see for instance [C˘al00], [C˘al02]. But this group vanishes as soon as the variety MX (2, 1, mg) is a nonsingular curve (see again [C˘al02]). Hence we have a universal vector bundle on X × MX (2, 1, mg). We consider a component M of MX (2, 1, mg) and we let E be the restriction of the universal sheaf to X × M. We let p and q be the projections of X × M respectively to X and M. We prove now the irreducibility of MX (2, 1, mg). Denoting by Ey the ∼ restriction of E to X × {y}, we have Ey = Ez if and only if y = z, for y, z ∈ M. Moreover, for any sheaf F in MX (2, 1, mg), we have: k ∼ ExtX (Ey,F ) = 0, for k = 2, 3, and for all k if F =6 Ey, where the vanishing for k = 2 follows from Lemma 3.3. Hence we have: k ∗ ∗ R q∗(E ⊗ p (F )) = 0, for k 6= 1, 1 ∗ ∗ ∼ ∼ R q∗(E ⊗ p (F )) = Oy, for F = Ey. 1 ∗ ∗ In particular, for any sheaf F in MX (2, 1, mg), the sheaf R q∗(E ⊗ p (F )) 1 ∗ ∗ has rank zero and we have χ(R q∗(E ⊗ p (F ))) = 1, for this value can be computed by Grothendieck-Riemann-Roch. Thus there must be a point 1 ∼ y ∈ M such that ExtX (Ey,F ) 6= 0, hence HomX (Ey,F ) 6= 0, so F = Ey. This implies that F belongs to M. Proposition 3.5. Let X be a smooth ordinary non-hyperelliptic prime Fano threefold of genus g and let F be a sheaf in MX (2, 1, mg). Then F is locally free and ACM. Furthermore, the moduli space MX (2, 1, mg) contains a sheaf F satisfying the conditions: 2 (3.2) ExtX (F,F ) = 0, ∼ ∼ (3.3) F ⊗ OL = OL ⊕ OL(1), for some line L with NL = OL ⊕ OL(−1).

Moreover, for each value of g, the following table summarizes the properties of F and MX (2, 1, mg). 3 g HX mg F g.g. The space MX (2, 1, mg) χ(F,F ) 0 3 4 3 6 ∃ H1 (X) 0 4 6 3 ∃ two points 1 5 8 4 ∃ Hesse septic curve 0 6 10 4 ∀ one point 1 7 12 5 ∀ smooth curve of genus 7 0 8 14 5 ∀ one point 1 9 16 6 ∀ smooth plane quartic 0 10 18 6 ∀ one point 1 12 22 7 ∀ one point 1

Proof. If g ≥ 6, the moduli space MX (2, 1, mg) is smooth by Lemma 3.3, and irreducible if g is even by the same lemma. If g is odd, irreducibility 10 MARIA CHIARA BRAMBILLA AND DANIELE FAENZI follows from Lemma 3.4. From Proposition 3.2 it follows that any sheaf F in MX (2, 1, mg) is a globally generated vector bundle which is ACM. The sheaf F in this case will also satisfy (3.3), indeed F has degree 1 and is globally generated on L. Note that (3.2) is equivalent to the moduli space MX (2, 1, mg) being smooth of expected dimension 1 − χ(F,F ) at the point [F ]. On the other hand, (3.3) holds if there exists a global section of F whose vanishing locus does not meet L. Let us now review in detail the cases 3 ≤ g ≤ 5. 4 If g = 3, the threefold X is a smooth quartic hypersurface in P , since X is not hyperelliptic. By [Mad00], the moduli space MX (2, 1, 3) is birational 0 to the Hilbert scheme H1 (X) of lines contained in X, which has a reduced component for X is ordinary. Indeed, a general plane Λ containing a line L ⊂ X deﬁnes a plane cubic C as the residual curve of L in X ∩ Λ. The curve C is thus projectively normal, and by Hartshorne-Serre’s construction (see Proposition 2.2 and Lemma 2.3), this provides an ACM vector bundle 0 F lying in MX (2, 1, 3). Choosing a general line L which does not meet C, we get (3.3). One can easily prove that F cannot be globally generated, for it has only three independent global sections. For g = 4, the threefold X must be the complete intersection of a quadric 5 Q and a cubic in P . The result follows from [Mad02, Section 3.2], and we only have to check (3.2). If the quadric Q is nonsingular, one considers the bundles F1 and F2 obtained restricting to X the two nonisomorphic spinor bundles S1 and S2 on Q. Note that F1 and F2 are not isomorphic, with k ExtX (Fi,Fi) = 0 for k ≥ 1. One can check this computing the vanishing of k ∗ H (Q, Si ⊗ Si (−3)) for all k, which in turn follows from Bott’s theorem. Thus the moduli space MX (2, 1, 4) consists of two reduced points. Each of the bundles Fi is globally generated, hence (3.3) follows. By Bott’s theorem each Si is ACM, hence each Fi is ACM too. We leave the remaining case to the reader, namely, when the rank of the quadric Q is 5. For g = 5, the threefold X is obtained as the complete intersection of 6 three quadrics in P . One considers thus the net generated by the three quadrics as a projective plane Π. The set of singular quadrics in this net is the so-called Hesse curve ∆ ⊂ Π. It is a plane septic with only ordinary double points, see for instance [Bea77]. Following [Mad02, Section 3.2], one can prove that the moduli space MX (2, 1, 4) is birational to the curve ∆, so (3.2) holds. More precisely, a singular quadric corresponding to a point of 3 ∆ contains a P which cuts on X a projectively normal elliptic quartic C. The Hartshorne-Serre correspondence thus provides an ACM vector bundle F . Since the ideal sheaf of C is generated by three linear forms, the vector bundle F is globally generated and (3.3) follows.

3.2. Moduli of ACM 2-bundles with higher c2. The main result of this section is the following existence theorem. Theorem 3.6. Let X be an ordinary smooth non-hyperelliptic prime Fano threefold of genus g. Then there exists an ACM vector bundle F of rank 2 with c1(F ) = 1 and c2(F ) = d for any mg ≤ d ≤ g + 3. The bundle F lies in a reduced component of dimension 2d − g − 2 of MX (2, 1, d). ACM BUNDLES ON PRIME FANO THREEFOLDS 11

We will need a series of lemmas. The following one is proved in [BF07, Theorem 3.7]. Lemma 3.7. Let X be as in Theorem 3.6 and let L be a general line in X. Then, for any integer d ≥ mg, there exists a rank 2 stable locally free sheaf F with c1(F ) = 1, c2(F ) = d, and satisfying: 2 (3.4) ExtX (F,F ) = 0, (3.5) H1(X,F (−1)) = 0, ∼ (3.6) F ⊗ OL = OL ⊕ OL(1), ∼ where L is a line with NL = OL ⊕ OL(−1). As shown in the proof of [BF07, Theorem 3.7], we recall that given a stable 2-bundle Fd−1 with c1(Fd−1) = 1 and c2(Fd−1) = d − 1, satisfying the properties (3.4), (3.5) and (3.6) and a general L line contained in X, we have the following exact sequence: σ (3.7) 0 → Sd → Fd−1 → OL → 0, where σ is the natural surjection and Sd = ker(σ) is a nonreﬂexive sheaf in MX (2, 1, d).

Definition 3.8. Let M(mg) be a component of MX (2, 1, mg) containing a stable locally free sheaf F satisfying the three conditions (3.4), (3.5) and (3.6). This exists by Proposition 3.5, and coincides with MX (2, 1, mg) for g ≥ 6. For each d ≥ mg + 1, we recursively define N(d) as the set of nonreflexive sheaves Sd fitting as kernel in an exact sequence of the form (3.7), with Fd−1 ∈ M(d − 1), and M(d) as the component of the moduli scheme MX (2, 1, d) containing N(d). We have: dim(M(d)) = 2 d − g − 2.

Lemma 3.9. For each mg ≤ d ≤ g + 2, the general element Fd of MX (d) satisﬁes: 0 h (X,Fd) = g + 3 − d, k H (X,Fd) = 0, for k ≥ 1. 3 Proof. Note that H (X,Fd) = 0 for all d by Serre duality and stability, and by Riemann-Roch we have χ(Fd) = g + 3 − d. We prove the remaining claim by induction on d. The ﬁrst step of the induction corresponds to d = mg, and it follows from Proposition 3.5. Assume now that the statement holds for Fd−1 with d ≤ g + 2, and let us prove it for a general element Sd of NX (d). By semicontinuity the claim will follow for the general element Fd ∈ M(d). So let Fd−1 be a locally free sheaf in 0 MX (d − 1). By induction we know that h (X,Fd) = g + 3 − d + 1 ≥ 2. A nonzero global section s of Fd−1 gives the exact sequence: s (3.8) 0 → OX −→ Fd−1 → IC (1) → 0, where C is a curve of degree d − 1 and arithmetic genus 1. We have 0 h (X, IC (1)) = g + 3 − d ≥ 1, so C is contained in some hyperplane section surface S given by a global section t of IC (1). Let L be a general line such ∼ that Fd−1 ⊗ OL = OL ⊕ OL(1) and L meets S at a single point x. We may 12 MARIA CHIARA BRAMBILLA AND DANIELE FAENZI assume the latter condition because there exists a line in X not contained in S. Indeed, the lines contained in X sweep a divisor of degree greater than one, see section 2.3. We can write down the following exact commutative diagram: 0 0 OX OX t t s s> 0 / OX / Fd−1 / IC (1) / 0 t> s 0 / OX / ID(1) / OS(HS − C) / 0 0 0 0 which in turn yields the exact sequence:

s 2 (t) 0 → OX −−→ Fd−1 → OS(HS − C) → 0, and dualizing we obtain:

> > ∗ (s t ) 2 (3.9) 0 → Fd−1 −−−−→ OX → OS(C) → 0. Thus the curve C moves in a pencil without base points in the surface S, and each member C0 of this pencil corresponds to a global section s0 of 0 Fd−1 which vanishes on C . Therefore we can choose s so that C does not ∼ contain x. In particular we get OL ⊗ IC (1) = OL(1). Now let σ be the natural surjection Fd−1 → OL and Sd = ker(σ). We have thus the exact 2 sequence (3.7). Taking global sections we obtain H (X, Sd) = 0. By tensoring (3.8) by OL we see that the composition σ ◦ s must be nonzero (in fact it is surjective). Thus the section s does not lift to Sd for 0 0 d ≤ g + 2, so h (X, Sd) ≤ h (X,Fd−1) − 1. We have thus: 0 0 0 h (Sd) ≥ χ(Sd) = χ(Fd−1) − 1 = h (Fd−1) − 1 ≥ h (Sd), and our claim follows. Using Lemma 2.3 we deduce the following corollary. Corollary 3.10. For d ≤ g + 2, let D be the zero locus of a nonzero global section of a general element F of MX (d). Then we have: 0 h (X, ID(1)) = g + 2 − d,

Lemma 3.11. For d ≥ g + 3, the general element Fd of MX (d) satisﬁes: 1 h (X,Fd) = d − g − 3, k H (X,Fd) = 0. for k 6= 1.

Proof. By the proof of Lemma (3.9), we can choose a sheaf Fd−1 ∈ M(d−1), a line L ⊂ X, a projection σ : Fd−1 → OL, and a global section s ∈ ACM BUNDLES ON PRIME FANO THREEFOLDS 13

0 H (X,Fd−1), Sd = ker(σ) such that σ ◦ s is surjective. Therefore we have the following exact diagram: (3.10) 0 0 0 / IL / OX / OL / 0 s σ 0 / Sd / Fd−1 / OL / 0 IC (1) IC (1) 0 0 0 ∼ The leftmost column induces, for all d, an isomorphism H (X, Sd) = 0 H (X, IC (1)). Assume now d = g + 3. Then Corollary 3.10 implies 0 H (X, Sg+3) = 0. Taking cohomology of the middle row of (3.10), by in- 2 3 duction we get H (X, Sg+3) = H (X, Sg+3) = 0. By semicontinuity, we k also have H (X,Fg+3) = 0, for k = 0, 2, 3. For higher d the claim is now obvious. Here is the proof of the main result of this section.

Proof of Theorem 3.6. We work by induction on d ≥ mg. Note that by Proposition 3.5, the statement holds for d = mg. Let F be a general sheaf in M(d) and recall that F is obtained as a general deformation of a sheaf Sd ﬁtting into an exact sequence of the form (3.7), where Fd−1 is a vector bundle in M(d). By induction we assume that Fd−1 is ACM. Let C be the zero locus of a nonzero global section of Fd−1. By the proof of Lemma 3.11 we have thus an exact sequence:

0 → IL → Sd → IC (1) → 0. 1 1 Since L is projectively normal, this implies H∗(X, Sd) ⊂ H∗(X, IC (1)). 1 ∼ 1 By Lemma 2.3 we have H∗(X, IC (1)) = H∗(X,Fd−1), and this module vanishes by the induction hypothesis. 1 So we obtain H∗(X, Sd) = 0, hence by semicontinuity the module 1 H∗(X,F ) is zero as well. Then by Serre duality the vector bundle F is ACM. 3.3. Elliptic and half-canonical curves. Choose a general element F of

M(mg), and consider the zero locus Cmg of a general global section of F . We can assume that Cmg is a smooth elliptic curve of degree m. For d ≥ mg + 1 1 1 1 we consider the subsets Kd , Ld of Hd (X) deﬁned as follows: 1 1 (3.11) Kmg = the component of Hmg (X) containing [Cmg ], 1 1 0 (3.12) Ld = {[C ∪ L] | len(C ∩ L) = 1, [C] ∈ Kd−1,L ∈ H1 (X)}, 1 1 1 (3.13) Kd = the component of Hd (X) containing Ld .

Lemma 3.12. For each mg ≤ d ≤ g+1, let Fd be a general element of M(d) and L be a general line contained in X. Then there exists a global section s 14 MARIA CHIARA BRAMBILLA AND DANIELE FAENZI of Fd whose zero locus C meets L at a single point and we have: 0 0 h (X, IC∪L(1)) = h (X, IC (1)) − 1. Proof. Reasoning as in the proof of the previous lemma, we choose a hyperplane section surface S containing the zero locus C of a given global section of Fd and meeting L at a single point x. Note that the line bundle OS(C) has a global section which vanishes 0 at the point x. Indeed by (3.9) we have h (S, OS(C)) = 2, which im- 0 plies h (S, Ix,S ⊗ OS(C)) = 1. The corresponding global section s of Fd satisﬁes our statement. Indeed, since L is not contained in S, we have 0 0 h (X, IC∪L(1)) ≤ h (X, IC (1)) − 1. So the claim follows taking global sections of the natural exact sequence:

0 → IC∪L(1) → IC (1) → OL → 0. Note that the sections satisfying the claim form a vector subspace of 0 H (X,Fd) of codimension 1. Lemma 3.13. Let D be the zero locus of a global section of a sheaf F lying 1 in MX (2, 1, d) satisfying (3.4), (3.5) and such that H (X, ID(1)) = 0. Then 2 we have ExtX (ID, ID) = 0 and we obtain an exact sequence: 0 1 1 (3.14) 0 → H (X, ID(1)) → ExtX (ID, ID) → ExtX (F,F ) → 0, 1 so extX (ID, ID) = d.

Proof. We apply the functor HomX (F, −) to the exact sequence (2.6). It is k easy to check that ExtX (F, OX ) = 0 for any k, thus we obtain for each k an isomorphism: k ∼ k ExtX (F, ID(1)) = ExtX (F,F ).

Therefore, applying HomX (−, ID(1)) to (2.6), we get the vanishing 2 ExtX (ID, ID) = 0 and, since F is a stable sheaf, we obtain the exact 1 sequence (3.14). The value of extX (ID, ID) can now be computed by Riemann-Roch. Proposition 3.14. Let X be an ordinary smooth non-hyperelliptic prime Fano threefold of genus g and assume X to be general for g ≤ 5. For 1 1 mg ≤ d ≤ g + 2, the subset Kd of Hd (X) is reduced of dimension d, and it contains a dense open subset consisting of smooth integral projectively 1 1 1 normal curves. The subset Ld of Hd (X) is a divisor in Kd . Proof. Let D be the zero locus of a nonzero global section of a vector bundle F in M(d). We can assume that F is ACM, so by Lemma 2.3 we have 1 H∗(X, ID) = 0 so D is projectively normal. We would like to prove that D can be chosen smooth integral. We work by induction on d ≥ mg. Set d = mg. For g ≥ 4, by Proposition 3.5 we can assume that F is globally generated so D is smooth integral if the global section of F is general. For g = 3, this was proved in [Mad00]. Assume thus that C is a smooth projectively normal curve, given by a section sd−1 of a vector bundle Fd−1 in M(d − 1) which satisﬁes (3.4). Suppose that sd−1 is a global section provided by Lemma 3.12. Then by ACM BUNDLES ON PRIME FANO THREEFOLDS 15 construction we have the commutative exact diagram: (3.15) 0 0 OX OX

s sd−1 d 0 / Sd / Fd−1 / OL / 0 0 / IC∪L(1) / IC (1) / OL / 0 0 0 1 By Lemma 3.12 we have H (X, IC∪L(1)) = 0 and by the proof of Theorem 2 1 3.7 we have ExtX (Sd, Sd) = 0. Thus the variety Kd is smooth of dimension 1 d at [C ∪ L] by Lemma 3.13. Moreover, the subvariety Ld has codimension 1 1 1 in Kd , i.e. dim(Ld ) = d − 1. Indeed, by Lemma 3.12, the sections sd−1 0 of Fd−1 which give rise to (3.15) form a vector subspace W0 ⊂ H (X,Fd−1) 0 0 of codimension 1. Projecting W0 ∩ H (X, Sd) to H (X, IC∪L(1)) via the map induced by (2.6) we obtain a subspace W of codimension 1. That is, 1 dim(W ) = g − d + 1. Thus the tangent space T to Ld at [C ∪ L] fits into an exact sequence analogous to (3.14) of the form: 1 0 → W → T → ExtX (Sd, Sd) → 0. 1 Since extX (Sd, Sd) = 2d − g − 2, and dim(W ) = g − d + 1, we conclude that T has dimension d − 1. We have thus proved that a general deformation D of the curve C ∪ L in 1 Kd cannot contain a line, hence by semicontinuity, it must be an integral curve. More than that, this curve must be nonsingular. Indeed, assume the contrary, and consider the open neighborhood [D] ∈ Ω consisting of smooth 1 points of Kd . For each point b ∈ Ω, consider the curve Cb, and its (unique) singular point xb ∈ Cb. Blowing up the trace of xb in X × Ω, we get a flat family of smooth rational curves C˜ in X˜, parametrised by Ω. But for 1 ˜ b ∈ Ω∩Ld the curve Cb is disconnected, a contradiction by [Har77, Chapter III, exercise 11.4]. Proposition 3.15. The general element of M(g+3) corresponds to a locally free sheaf F such that F (1) is globally generated and ACM with c1(F (1)) = 3, c2(F (1)) = 5 g − 1. In particular, a general global section of F (1) vanishes along a smooth half-canonical curve of genus 5 g. Proof. By Lemma 3.11, F satisfies H0(X,F ) = H1(X,F ) = 0. By stability we have H3(X,F ) = 0 so Riemann-Roch implies H2(X,F ) = 0. If the sheaf F is general, we have thus: h1(X,F ) = 0, h2(X,F (−1)) = h1(X,F (−1)) = 0 by (3.5), h3(X,F (−2)) = h0(X,F ) = 0.

So F (1) is globally generated by Castelnuovo-Mumford regularity. 16 MARIA CHIARA BRAMBILLA AND DANIELE FAENZI

4. Bundles with even first Chern class In this section we will prove the existence of the remaining case of Madonna’s list, namely case (d) of Remark 2.5, see Theorem 4.10. We will µ need a preliminary analysis of the moduli space MX (2, 0, 2), which turns out 0 to be in bijection with H2 (X). The investigation of the moduli space of ACM bundles with c1 = 0, c2 = 4 is carried out via deformations in section 4.2. Finally, we show some further properties of MX (2, 0, 4) for threefolds of high genus.

4.1. Conics and 2-bundles with c2 = 2. Here we study rank 2 sheaves on 0 X with c1 = 0, c2 = 2, and their relation with the Hilbert scheme H2 (X) of conics contained in X. We rely on well-known properties of the Hilbert 0 scheme H2 (X), see section 2.3. Lemma 4.1. Let X be a smooth non-hyperelliptic prime Fano threefold of genus g. Then any Cohen-Macaulay curve C of degree 2 has pa(C) ≤ 0. Moreover if C is nonreduced it must be a Gorenstein double structure on a line L deﬁned by the exact sequence:

(4.1) 0 → IC → IL → OL(t) → 0, ∼ where t ≥ −1 and we have pa(C) = −1 − t and ωC = OC (−2 − t).

Proof. If C is reduced, clearly it must be a conic (then pa(C) = 0) or the union of two skew lines (then pa(C) = −1). So assume that C is nonreduced, hence a double structure on a line L. By [Man86, Lemma 2] we have the exact sequences: 2 2 0 → IC /IL → IL/IL → OL(t) → 0, (4.2) 0 → OL(t) → OC → OL → 0, and C is a Gorenstein structure given by Ferrand’s doubling (see [Man94], [BF81]). 2 ∼ ∗ ∗ ∼ Recall that IL/IL = NL. By [Isk78, Lemma 3.2] we have either NL = ∗ ∼ OL ⊕ OL(1), or NL = OL(−1) ⊕ OL(2). It follows that t ≥ −1 and we obtain (4.1). We compute that c3(IL) = −1 and c3(OL(t)) = 1 + 2t, hence c3(IC ) = −2 − 2t, so pa(C) = −1 − t. Dualizing (4.1), by the fundamental local isomorphism we obtain the exact sequence:

(4.3) 0 → OL(−2) → ωC → OL(−2 − t) → 0, which by functoriality is obtained twisting by OX (−2−t) the exact sequence (4.2). This concludes the proof. Corollary 4.2. Let X a smooth non-hyperelliptic prime Fano threefold. 1 Then any conic contained in X is reduced if and only if H0 (X) is smooth. This takes place if X is general. 1 Proof. From [IP99, Proposition 4.2.2], the Hilbert scheme H0 (X) is smooth ∼ if and only if we have NL = OL⊕OL(−1) for any line L in X. By the previous lemma this is equivalent to the fact that any conic contained in X is reduced. 1 Notice that if X is general, by [IP99, Theorem 4.2.7], we have that H0 (X) is smooth. ACM BUNDLES ON PRIME FANO THREEFOLDS 17

We will need the following lemma. Lemma 4.3. Let X be a smooth prime non-hyperelliptic Fano threefold of genus g and F be a µ-semistable sheaf in MX (2, 0, c2, c3), with c2 < mg. Then we have: (4.4) H2(X,F ) = 0, (4.5) H2(X,F (1)) = 0.

Proof. Let us ﬁrst prove (4.5). Assume by contradiction that H2(X,F (1)) 6= 1 0. Thus a nonzero element of Ext (F, OX (−2)) provides an exact sequence of the form:

(4.6) 0 → OX (−2) → F˜ → F → 0, where the sheaf F˜ has rank 3 and Chern classes c1(F˜) = −2, c2(F˜) = c2. The sheaf F˜ cannot be semistable by Bogomolov’s inequality, see (2.3). Thus we consider the Harder-Narasimhan ﬁltration 0 ⊂ K1 ⊂ · · · ⊂ F˜, which provides the following exact sequence:

0 → K1 → F˜ → Q → 0, where the semistable sheaf K1 destabilizes F˜, and the sheaf Q is torsionfree. Consider first the case rk(K1) = 2. This implies either c1(K1) = 0 or c1(K1) = −1. In the former case, the composition K1 ,→ F˜ → F must be injective, for its kernel would be a degree-zero subsheaf of OX (−2). This implies that OX (−2) sits into Q, which is a rank 1 torsionfree sheaf with ∼ ∼ c1(Q) = −2. It follows that Q = OX (−2) and so we have K1 = F and the sequence (4.6) splits, a contradiction. In the latter case, we have c2(K1) ≥ mg since K1 and its restriction to a general hyperplane section must satisfy (2.4) by Maruyama’s theorem [Mar80]. On the other hand, Q is a rank 1 torsionfree sheaf hence c2(Q) ≥ 0. Since c1(Q) = −1, we get mg > c2(F˜) = c2(K1) + c2(Q) + 2g − 2 ≥ 2g − 2 + mg, a contradiction. Consider now the case rk(K1) = 1. Note that c1(K1) equals 0 and c2(K1) is nonnegative. It follows that rk(Q) = 2, c1(Q) = −2 and c2(Q) ≤ c2 < mg. Notice that Q is not µ-semistable Bogomolov’s inequality. Then, a Harder-Narasimhan filtration of Q gives the exact sequence: 0 → P → Q → R → 0, where P,R are rank 1 torsionfree sheaves with nonnegative c2, and c1(P ) ≥ 0. Notice that (4.6) induces an exact diagram: 0 0 K1 K1 0 / OX (−2) / F˜ / F / 0 0 / OX (−2) / Q / M / 0 0 0 18 MARIA CHIARA BRAMBILLA AND DANIELE FAENZI which in turn induces an injective map either OX (−2) ,→ P , or OX (−2) ,→ R. In the former case, we obtain a surjective map M → R, hence R is a destabilizing quotient for F , contradiction. In the latter case, since c1(R) ≤ ∼ −2, we have R = OX (−2). Then Q contains OX (−2) as direct summand, so (4.6) splits, a contradiction. This finishes the proof of (4.5). 1 We shall now prove H (S, FS(1)) = 0, where S is a general hyperplane section surface. Note that, together with (4.5), this easily implies (4.4). Assume by contradiction that H1(S, F (1)) 6= 0. By Serre duality, this provides a nontrivial extension of the form:

(4.7) 0 → OS(−1) → FfS → FS → 0, where the sheaf FfS has rank 3 and Chern classes c1(FfS) = −1, c2(FfS) = c2 < mg. Note that FfS cannot be stable, for the space MS(3, −1, c2) is empty by the dimension count (2.4). Then, the Jordan-H¨older ﬁltration gives:

0 → K1 → FfS → Q → 0, where this time the destabilizing sheaf K1 and the torsionfree sheaf Q are defined on S. To show that this leads to a contradiction, we repeat the above argument, where OX (−2) is replaced with OS(−1). The only difference is that, if the destabilising subsheaf K1 of FfS has rank 1, then the quotient Q = FfS/K1 has rank 2 and c1(Q) = −1, c2(Q) < mg. Thus Q is not stable by the dimension count (2.4). Taking again a Harder-Narasimhan filtration of Q as above we can show that the extension (4.7) should split, which is again a contradiction.

Remark 4.4. Note that in the previous lemma, the hypothesis c2(F ) < mg cannot be dropped. In fact, Iliev and Markushevich in [IM00b, Proposition 1.4, arXiv version] proved that there exists a component of MX (2, 0, 4), where a general element F satisﬁes h1(X,F (1)) = 1, and this contradicts 1 2 either H (S, FS(1)) = 0 or H (X,F ) = 0. Remark 4.5. Let X be as in the previous lemma, and let P a torsionfree sheaf of rank 1 with c1(P ) = 0, c2(P ) ≤ mg. Taking F = OX ⊕ P in Lemma 4.3, we get H2(X,P ) = 0. Note that, unless P is isomorphic to 0 OX , we also have H (X,P ) = 0, hence, χ(P ) must be nonpositive. Now, by 2−c2(P )+c3(P ) Riemann-Roch, we have χ(P ) = 2 . This implies:

(4.8) c3(P ) ≤ c2(P ) − 2. Proposition 4.6. Let X be a smooth non-hyperelliptic prime Fano threefold. Then the moduli space MX (2, 0, 2) is empty. Moreover any sheaf F in µ µ MX (2, 0, 2) is strictly µ-semistable, and the moduli space MX (2, 0, 2) is in 0 bijection with the Hilbert scheme H2 (X). Proof. By Lemma 4.3, we have H2(X,F ) = 0. Since χ(F ) = 1, we ob- 0 tain H (X,F ) 6= 0. So OX is a destabilizing subsheaf of F in the sense of µ Gieseker. Therefore, an element F of MX (2, 0, 2) must be strictly semistable. A Jordan-H¨olderﬁltration of F amounts to an exact sequence: (4.9) 0 → P → F → R → 0, ACM BUNDLES ON PRIME FANO THREEFOLDS 19 where P,R are torsionfree sheaves of rank 1. The graded associated to F is ∼ ∼ gr(F ) = P ⊕ R, and we want to prove that gr(F ) = OX ⊕ IC for a conic 0 C ∈ H2 (X). By semistability of F we have:

c1(P ) = c1(R) = 0, c2(P ) + c2(R) = 2, c3(P ) = −c3(R), and since P is torsionfree, we must have c2(P ) ≥ 0 and by the same reason c2(R) ≥ 0. Summing up, we must take into account the three possibilities: c2(P ) = 2, c2(P ) = 1 or c2(P ) = 0. Note that the third case is equivalent to the ﬁrst one. In the ﬁrst case we have c2(R) = 0, and thus we have c3(R) ≤ 0 and c3(P ) ≥ 0. Then the singular locus of P contains a Cohen-Macaulay curve C of degree 2, so we have an exact sequence

0 → P → IC → Q → 0, where Q is a torsion sheaf with c1(Q) = c2(Q) = 0 and c3(Q) ≥ 0. On the other hand c3(IC ) = 2pa(C) ≤ 0 by Lemma 4.1. Then c3(P ) = c3(IC ) − c3(Q) ≤ 0. Hence we obtain c3(P ) = 0, moreover we get c3(Q) = 0 and ∼ ∼ P = IC where pa(C) = 0. Finally since c3(R) = 0 we get R = OX . It ∼ follows gr(F ) = IC ⊕ OX . In the second case, since c2(P ) = 1, the singular locus of P contains a line and the quotient Q = IL/P is a torsion sheaf with c1(Q) = c2(Q) = 0 and c3(Q) ≥ 0. Hence c3(P ) = −1 − c3(Q) < 0. Analogously since c2(R) = 1, we get c3(R) < 0, a contradiction. Remark 4.7. Note that the previous argument shows that a rank 1 torsionfree sheaf P with c1(P ) = 0, c2(P ) = 2 satisﬁes c3(P ) ≤ 0. One proves analogously that if c2(P ) = 1, then c3(P ) ≤ −1. µ Lemma 4.8. Given X as above, any element in MX (2, 0, 2) is represented by a locally free sheaf F C satisfying h0(X,F C ) = 1, Hk(X,F C ) = 0 for k ≥ 1. µ Proof. Given a class [F ] in MX (2, 0, 2), by Proposition 4.6, gr(F ) is isomorphic to OX ⊕ IC , for some conic C contained in X. Note that C is subcanonical: this is clear if it is reduced, and it follows by Lemma 4.1 otherwise. By the Hartshorne-Serre correspondance, we get that F is equivalent to an extension of the form: C (4.10) 0 → OX → F → IC → 0, C k and F is locally free. Note that H (X, IC ) = 0 for all k: this follows by C (4.1) if C is nonreduced. Thus the cohomology of F behaves as desired. The following lemma will be needed later on.

Lemma 4.9. Given X as above, let F be a locally free sheaf in MX (2, 0, 2 d), ∼ 2 C be a smooth conic contained in X with normal bundle NC = OC , and x ∼ 2 2 be a point of C. Assume that F ⊗ OC = OC and that ExtX (F,F ) = 0. Let S be a sheaf ﬁtting into an exact sequence of the form:

(4.11) 0 → S → F → OC → 0. 0 2 Then we have H (C, S (−x)) = 0 and ExtX (S , S ) = 0. 20 MARIA CHIARA BRAMBILLA AND DANIELE FAENZI

0 Proof. To prove the vanishing of H (C, S (−x)), we tensor (4.11) by OC and we get the following exact sequence of sheaves on C: X (4.12) 0 → T or1 (OC , OC ) → S ⊗ OC → F ⊗ OC → OC → 0. X ∗ ∼ Recall that T or1 (OC , OC ) is isomorphic to NC = OC ⊕ OC . Now, twisting (4.12) by OC (−x) and taking global sections, we easily get H0(C, S (−x)) = 0. 2 Now let us prove the vanishing of ExtX (S , S ). Applying the functor HomX (−, S ) to (4.11) we obtain the exact sequence: 2 2 3 ExtX (F, S ) → ExtX (S , S ) → ExtX (OC , S ). We will prove that both the ﬁrst and the last terms of the above sequence vanish. Consider the former, and apply HomX (F, −) to (4.11). We get the exact sequence: 1 2 2 ExtX (F, OC ) → ExtX (F, S ) → ExtX (F,F ). 2 1 ∼ 1 By assumption we have ExtX (F,F ) = 0 and ExtX (F, OC ) = H (C,F ) = 2 0. We obtain ExtX (F, S ) = 0. To show the vanishing of the group 3 ExtX (OC , S ), we apply Serre duality, obtaining: 3 ∗ ∼ ∼ 0 ExtX (OC , S ) = HomX (S , OC (−1)) = H (X, H omX (S , OC (−1))).

To show that this group is zero, apply the functor H omX (−, OC ) to the sequence (4.11) to get: ∗ 0 → OC → F ⊗ OC → H omX (S , OC ) → NC → 0, ∼ 3 which implies H omX (S , OC (−1)) = OC (−1) , and this sheaf has no global sections.

4.2. Rank 2 bundles with c2 = 4. In this section, we begin the study of semistable sheaves F with Chern classes c1(F ) = 0, c2(F ) = 4, c3(F ) = 0 on smooth non-hyperelliptic prime Fano threefolds, and we prove the existence of case (d) of Madonna’s list. The main result of this part is the following. Theorem 4.10. Let X be a smooth non-hyperelliptic prime Fano threefold. Then there exists a rank 2 ACM stable locally free sheaf F with c1(F ) = 0, c2(F ) = 4. The bundle F lies in a reduced component of dimension 5 of the space MX (2, 0, 4). We will need the following lemma. Lemma 4.11. Let X as above and let C and D be smooth disjoint conics contained in X with trivial normal bundle. Then a sheaf S2 ﬁtting into a nontrivial extension of the form:

(4.13) 0 → IC → S2 → ID → 0, is simple.

Proof. In order to prove the simplicity, apply HomX (S2, −) to (4.13) and get

HomX (S2, IC ) → HomX (S2, S2) → HomX (S2, ID). ACM BUNDLES ON PRIME FANO THREEFOLDS 21

The ﬁrst term vanishes, indeed applying HomX (−, IC ) to (4.13) and since C ∩ D = ∅ we get

δ 1 0 → HomX (S2, IC ) → HomX (IC , IC ) → ExtX (ID, IC ). Clearly the map δ : C → C is nonzero, hence HomX (S2, IC ) = 0. On the other hand, applying HomX (−, ID) to (4.13) we get: ∼ ∼ HomX (S2, ID) = HomX (ID, ID) = C, and we deduce homX (S2, S2) = 1, i.e. the sheaf S2 is simple. Proof of Theorem 4.10. Choose two smooth conics C and D in X with trivial normal bundle. This is possible in view of well know results, see section 2.3. Let us check that we can assume that C and D are disjoint. Let S be a hyperplane section surface containing C. A general conic D intersects S at 2 points. Since X is covered by conics, moving D in 0 H2 (X), these two points sweep out S. Thus, a general conic D meets C at 0 most at a single point. This gives a rational map ϕ : H2 (X) → C. Note 0 that, for any point x ∈ C, we have H (C,NC (−x)) = 0. So there are only finitely many conics contained in X through x, for this space parametrises the deformations of C which pass through x. Thus the general fiber of ϕ is finite, which is a contradiction. D µ Let F be the vector bundle in MX (2, 0, 2) fitting into: ι D (4.14) 0 → OX → F → ID → 0. 2 D D One can easily prove the vanishing ExtX (F ,F ) = 0, since the normal bundle to D is trivial. Tensoring by OC the exact sequence (4.14), we obtain D ∼ 2 D FC = OC . Then we have homX (F , OD) = 2, and for any morphism D D D FC → OC we denote by σ the surjective composition σ : F → FC → OC . We can choose σ such that the composition σ ◦ ι : OX → OC is nonzero, i.e. such that:

(4.15) ker(σ) 6⊃ Im(ι) ⊗ OC .

We denote by S2 the kernel of σ and we have the exact sequence: D σ (4.16) 0 → S2 → F → OC → 0, and S2 ﬁts into (4.13). Clearly, the sheaf S2 represents an element of k MX (2, 0, 4), and by (4.13) we get H (X, S2) = 0 for all k. More than that, since a smooth conic is projectively normal, again by (4.13) we obtain: 1 (4.17) H∗(X, S2) = 0.

Note that the sheaf S2 is strictly semistable, and simple by Lemma 4.11. Now we would like to ﬂatly deform S2 to a stable ACM vector bundle F . We perform this deformation in the moduli space of simple sheaves SplX (2, 0, 4), see section 2.2. 2 Note that Lemma 4.9 gives ExtX (S2, S2) = 0. Hence by Riemann-Roch 1 and by the simplicity of S2 we get that ext (S2, S2) = 5. This implies that SplX (2, 0, 4) is smooth and locally of dimension 5 around the point [S2].

Claim 4.12. The set of sheaves in SplX (2, 0, 4) ﬁtting into an exact sequence of the form (4.16) forms a subset of codimension 1 in SplX (2, 0, 4). 22 MARIA CHIARA BRAMBILLA AND DANIELE FAENZI

Let us postpone the proof of the above claim, and assume it from now on. We may thus choose a deformation G of S2 in SplX (2, 0, 4) which does not ﬁt into (4.16), and, setting E = G∗∗, we write the double dual sequence: (4.18) 0 → G → E → T → 0. 1 By semicontinuity, we may assume homX (G, G) = 1, and H (X,G(−1)) = 0. We may also assume that, for any given line L contained in X, we 1 have the vanishing ExtX (OL(t),G) = 0 for all t ∈ Z. Indeed, applying HomX (OL(t), −) to (4.16) we get: 1 1 D HomX (OL(t), OC ) → ExtX (OL(t), S2) → ExtX (OL(t),F ), and observe that the leftmost term vanishes as soon as L is not contained in C (but C is irreducible), while the rightmost does for F D is locally free. Clearly E is a semistable sheaf, so H1(X,G(−1)) = 0 implies H0(X,T (−1)) = 0, hence T must be a pure sheaf supported on a Cohen- Macaulay curve B ⊂ X. Summing up, we have c1(T ) = 0 and c2(T ) < 0.

Claim 4.13. We may assume that c2(T ) ≥ −2. Hence in particular we have −c2(T ) ∈ {1, 2}. We postpone the proof of the last claim and we prove now that B must be ∼ a smooth conic. For otherwise, either T = ON (a) for some line L1 ⊂ X and some a1 ∈ Z (if c2(T ) = −1), or, if c2(T ) = −2 in view of Lemma 4.1, there must be a second line L2 ⊂ X (possibly coincident with L1), and a2 ∈ Z such that T ﬁts into:

(4.19) 0 → OL1 (a1) → T → OL2 (a2) → 0. 1 But we have seen that ExtX (OL(t), S2) = 0 for all t ∈ Z, and for any 1 line L ⊂ X. By semicontinuity we get ExtX (OL(t),G) = 0, for all t ∈ Z 1 and any line L ⊂ X. In particular ExtX (OLi (ai),G) = 0, for i = 1, 2, so 1 ExtX (T,G) = 0 and (4.18) should split, contradicting semistability of E. Therefore T must be of the form OB(a x), for some integer a, and for some point x of a smooth conic B ⊂ X. By [Har80, Proposition 2.6], we 0 have c3(E) = c3(T ) = 2 a ≥ 0, while H (X,T (−1)) = 0 implies a − 2 < 0. We are left with the cases a = 0 or a = 1. We may exclude the former by 1 Claim 4.12. The latter can be excluded too, by proving that ExtX (OB(x),G) is zero for any conic B ⊂ X. That this is indeed the case can be checked by semicontinuity applying HomX (OB(x), −) to (4.16), obtaining: 1 1 D HomX (OB(x), OC ) → ExtX (OB(x), S2) → ExtX (OB(x),F ). The rightmost term in the above sequence vanishes because F D is locally 0 free. The leftmost term is isomorphic to H (X, H om(OB(x), OC )), and the sheaf H om(OB(x), OC ) is zero for B 6= C. On the other hand, if B = C we ∼ 0 have HomX (OC (x), OC ) = H (C, OC (−x)) = 0. Summing up, we have proved that T must be zero, so G is isomorphic to E, and thus locally free. Since H0(X,G) = 0, the sheaf G must be stable. 1 By (4.17) and semicontinuity H∗(X,G(t)) = 0, so by Serre duality we get that G is ACM.

Proof of Claim 4.12. We have proved that a sheaf S2 ﬁtting into an exact sequence of the form (4.16), for some disjoint conics C,D ⊂ X, ﬁts also into ACM BUNDLES ON PRIME FANO THREEFOLDS 23

(4.13). So, we need only prove that the set of sheaves fitting into (4.13) is a closed subset of dimension 4 of SplX (2, 0, 4). Since C and D belong to the 0 surface H2 (X), it is enough to prove that there is in fact a unique (up to 1 isomorphism) nontrivial such extension, i.e. extX (ID, IC ) = 1. 3 Note that HomX (ID, IC ) = ExtX (ID, IC ) = 0, hence by Riemann-Roch 2 it suffices to prove ExtX (ID, IC ) = 0. Applying HomX (−, IC ) to the exact sequence defining D in X, we are reduced to prove the vanishing of 3 ExtX (OD, IC ). But this group is dual to: ∼ 0 ∼ 0 HomX (IC , OD(−1)) = H (X, H omX (IC , OD(−1))) = H (X, OC (−1)) = 0. Proof of Claim 4.13. We have already proved that H0(X,T (−1)) = 0 and this implies that χ(T (t)) = − h1(X,T (t)) for any negative integer t. Recall that, by [Har80, Remark 2.5.1], the reflexive sheaf E satisfies 1 H (X,E(t)) = 0 for all t 0. Thus, tensoring by OX (t) the exact sequence (4.18) and taking cohomology, we obtain h1(X,T (t)) ≤ h2(X,G(t)) for all t 0. Further, for any integer t, we can easily compute the following Chern classes: c1(T (t)) = 0, c2(T (t)) = c2(T ) = 4 − c2(E), c3(T (t)) = c3(E) − 2tc2(T ), hence by Riemann-Roch formula we have 1 χ(T (t)) = −tc (T ) + (c (E) − c (T )). 2 2 3 2 Since G is a general deformation of the sheaf S2, we also have, by semi- 2 2 continuity, h (X,G(t)) ≤ h (X, S2(t)). On the other hand, by (4.16) we 2 1 have h (X, S2(t)) = h (X, OC (t)) = −χ(OC (t)) = −2t−1. Summing up we have, for all t 0, the following inequality: 1 −tc (T ) + (c (E) − c (T )) ≥ 2t + 1, 2 2 3 2 which implies c2(T ) ≥ −2. The following result provides the existence of a well-behaved component of MX (2, 0, 2d), for each d ≥ 2. Theorem 4.14. Let X be a smooth non-hyperelliptic prime Fano threefold. Let C be a general conic in X, and let x be a point of C. For any integer d ≥ 2, there exists a rank 2 stable locally free sheaf Fd with c1(Fd) = 0, c2(Fd) = 2d, and satisfying: 2 (4.20) ExtX (Fd,Fd) = 0, 1 (4.21) H (X,Fd(−1)) = 0, 0 (4.22) H (C,Fd(−x)) = 0.

The bundle Fd lies in a reduced component of dimension 4d − 3 of the space MX (2, 0, 2d). Proof. We work by induction on d ≥ 2. In Theorem 4.10 we constructed a stable vector bundle as a deformation of a sheaf S2. We denote it by F2. By Lemma 4.9, the sheaf S2 satisﬁes (4.20), (4.22) and by the proof Theorem 4.10, it also satisﬁes (4.21). The same hols for F2 by semicontinuity. 24 MARIA CHIARA BRAMBILLA AND DANIELE FAENZI

By induction we can choose a stable locally free sheaf Fd−1 in MX (2, 0, 2d− 2), satisfying conditions (4.20), (4.22) and (4.21). Choose a smooth conic ∼ ∼ 2 C ⊂ X with NC = OC ⊕ OC . By (4.22) we deduce that Fd−1 ⊗ OC = OC , then we get hom(Fd−1, OC ) = 2. To any surjective morphism σ : Fd−1 → Fd−1 ⊗ OC → OC we associate the kernel Sd and we have the exact sequence: σ (4.23) 0 → Sd → Fd−1 → OC → 0.

Clearly Sd is a stable torsionfree sheaf with rank 2 and Chern classes c1(Sd) = 0 and c2(Sd) = 2d. In order to prove properties (4.20) and (4.22) we apply Lemma 4.9, while property (4.21) follows immediately from (4.23) and from the inductive hypothesis. Now, we would like to flatly deform the sheaf Sd to a stable vector bundle Fd. Notice that the set of sheaves in MX (2, 0, 2d) fitting into an exact sequence of the form (4.23) is a closed subset of dimension 4 d−4. Indeed by Riemann-Roch we know that the irreducible component M of MX (2, 0, 2d−2) containing [Fd−1] is smooth at [Fd−1] and has locally dimension 4 d − 7 0 and recall that H2 (X) is a surface. Since χ(Fd−1, OC ) = 2, the fact that 0 homX (Fd−1, OC ) = 2 takes place for general [C] ∈ H2 (X) and for general 0 [Fd−1] ∈ M. Hence given a general element ([Fd−1],C) ∈ M×H2 (X), to any map in Hom(Fd−1, OC ) it corresponds a sheaf Sd lying in MX (2, 0, 2d). This correspondence is injective by simplicity of Fd−1, hence the dimension of the set of sheaves in MX (2, 0, 2d) fitting in (4.23) has dimension 4 d−7+2+1 = 4 d − 4. On the other hand we have that the irreducible component of MX (2, 0, 2d) containing [Sd] is smooth at [Sd] and has locally dimension 4 d − 3. There- fore we may choose a deformation Fd of Sd which does not fit into (4.23), and we write the double dual sequence: ∗∗ 0 → Fd → Fd → T → 0. Following the argument used in the proof of Theorem 4.10, we conclude that T must be zero, and so F is locally free.

4.3. The space MX (2, 0, 4) on threefolds of high genus. Here we will give a more detailed description of the moduli space of rank 2 ACM bundles with c1 = 0 and c2 = 4 on threefolds of high genus. The ﬁrst result relates the existence of the bundle provided by Theorem 4.10 with that of a smooth canonical curve contained in X, when g is at least 5. The second one classiﬁes the sheaves in MX (2, 0, 4) for g at least 7. Proposition 4.15. Let X be a smooth non-hyperelliptic prime Fano threefold of genus g ≥ 5, and let F be a general element of the component of MX (2, 0, 4) given by Theorem 4.14. Then F (1) is globally generated. In particular, a general section of F (1) vanishes along a smooth projectively normal canonical curve of genus g + 2 and degree 2g + 2.

Proof. We prove that the sheaf S2(1) given by (4.16) is globally generated. Note that this implies that a general element in our component of MX (2, 0, 4) is globally generated. Indeed, let Ω be an open subset of

MX (2, 0, 4), and let F be a universal sheaf over X × Ω with [Fb0 ] = [S2]. ACM BUNDLES ON PRIME FANO THREEFOLDS 25

0 0 Observe that h (X, Fb(1)) = h (X, S2(1)) = 2 g − 2 for b general in Ω, since k H (X, S2(1)) = 0 for k ≥ 1 by (4.13). Then the natural evaluation map 2 g−2 OX×Ω → F(1) is surjective at b0, hence at a general point b. By the exact sequence (4.13), it is enough to prove that, for any conic C ⊂ X, the sheaf IC (1) is globally generated. Following Iliev-Manivel [IM07a, Proposition 5.4], see also [BF07, Proposition 3.4], one reduces to show that IC∩S(1) is globally generated on S for any smooth hyperplane section S of ∼ X with Pic(S) = Z. 1 The subscheme Z = C ∩ S has length 2, and note that H (S, IZ (1)) = 0. Assuming now that IZ (1) is not globally generated, we get a subscheme 0 0 0 Z ⊃ Z of length 3 and h (S, IZ0 (1)) = h (S, IZ (1)) = g − 1, therefore 1 h (S, IZ0 (1)) = 1, a contradiction by [BF07, Lemma 3.3] for g ≥ 5. Our result now follows by a Chern class computation since a general deformation of S2 is an ACM vector bundle.

In the following proposition, we give a classification of sheaves lying in MX (2, 0, 4), for threefolds of genus at least 7. Proposition 4.16. Let X be a smooth non-hyperelliptic prime Fano threefold of genus g ≥ 7 and F be a rank 2 semistable sheaf with c1(F ) = 0, c2(F ) = 4. Then the sheaf F is stable unless it fits into an exact sequence of the form 0 (4.13), with [C], [D] ∈ H2(X). If F is stable, then F is locally free unless it fits into one of the following exact sequences:

0 0 (4.24) 0 → IC → F → IL → 0, with [C] ∈ H3(X) and [L] ∈ H1(X), 0 (4.25) 0 → IC → F → Ix → 0, with [C] ∈ H4(X) and x ∈ C, −1 (4.26) 0 → IC → F → OX → 0, with [C] ∈ H4 (X), Proof. Assume ﬁrst that the sheaf F is strictly semistable, and consider its Jordan-Holder ﬁltration. This amounts to an exact sequence of the form (4.9), where P and R lie in M(1, 0, 2, 0). So P and R must be isomorphic to the ideal sheaf of a conic contained in X. We may thus assume that the sheaf F is stable, and let us assume that F is not locally free. Consider the double dual exact sequence:

0 → F → F ∗∗ → T → 0, where T is a torsion sheaf supported on a subscheme of codimension at ∗∗ ∗∗ least 2. We have c2(T ) ≤ 0 so c2(F ) ≤ 4. Note that the sheaf F is ∗∗ µ-semistable, hence c2(F ) ≥ 0 by Bogomolov’s inequality. Since g ≥ 7 implies mg ≥ 5, we may apply Lemma 4.3, to obtain 2 ∗∗ ∗∗ ∗∗ H (X,F ) = 0. On the other hand c3(F ) ≥ 0 gives χ(F ) ≥ 0, so H0(X,F ∗∗) 6= 0. We have thus an exact sequence:

∗∗ (4.27) 0 → OX → F → Q → 0.

∗∗ ∗∗ Assume ﬁrst c2(F ) > 0. Then F is not semistable, since OX is a destabilizing subsheaf, and we may consider the Harder-Narasimhan ﬁltration for 26 MARIA CHIARA BRAMBILLA AND DANIELE FAENZI

F ∗∗. This induces an exact commutative diagram: (4.28) 0 / P 0 / F / R0 / 0 0 / P / F ∗∗ / R / 0, where the vertical arrows are injective, and we have rk(P 0) = rk(R0) = 0 0 rk(P ) = rk(R) = 1 and c1(P ) = c1(R ) = c1(P ) = c1(R) = 0. We obtain the inequalities: c (F ) c (F ∗∗) (4.29) 2 = 2 ≥ c (R0) ≥ c (R) ≥ 2 > 0, 2 2 2 2 0 0 and c2(R ) = 2 forces c3(R ) > 0, which is impossible by Remark 4.7. It 0 remains to exclude c2(R ) = c2(R) = 1. Note that this implies c2(P ) = 1. ∗∗ By Remark 4.7 we have c3(R) ≤ −1, and c3(P ) ≤ −1. Since 0 ≤ c3(F ) = c3(P ) + c3(R), we get a contradiction. ∗∗ ∗∗ ∼ We have thus proved c2(F ) = 0. Let us show that this implies F = 2 OX . Indeed, taking the double dual of (4.27), we may assume that Q is ∗∗ torsionfree. Then c2(Q) = 0 and we have 0 ≥ c3(Q) = c3(F ) ≥ 0. One ∼ ∗∗ ∼ 2 gets Q = OX and F = OX . We may consider thus a diagram of the form (4.28), where this time 0 P = R = OX , and c2(R ) is either 0 or 1. 0 0 0 0 Now, if c2(R ) is zero, then c3(R ) ≤ 0. So c2(P ) = 4 and c3(P ) ≥ 0. 0 0 Applying (4.8) to P , we obtain c3(P ) = 2 or 0. These cases correspond respectively to (4.25) and (4.26). 0 0 0 On the other hand, in case c2(R ) = 1, then c3(R ) ≤ −1. Hence c2(P ) = 0 0 3 and c3(P ) ≥ 1. This time (4.8) implies c3(P ) = 1, i.e. case (4.24). Finally, let us see that, in case (4.25), the point x must lie in C. Ob- ∼ serve that otherwise one easily gets the isomorphism H omX (Ix, IC ) = IC , 1 2 and E xtX (Ix, IC ) = 0. Since H (X, IC ) = 0, by Riemann-Roch one ob- 1 1 tains H (X, IC ) = 0, so that the extension group ExtX (Ix, IC ) vanishes (by the local-to-global spectral sequence). Hence the sheaf F would not be semistable. Remark 4.17. By the previous proposition, the open subset of stable s sheaves MX (2, 0, 4) of the moduli space MX (2, 0, 4) consists of two disjoint `f s s `f subschemes MX (2, 0, 4) and NX (2, 0, 4) = MX (2, 0, 4) \ MX (2, 0, 4), where `f MX (2, 0, 4) is the subset of MX (2, 0, 4) consisting of locally free sheaves, s ∗∗ ∼ 2 while any sheaf F in NX (2, 0, 4) satisﬁes F = OX . s The scheme NX (2, 0, 4) contains a subset consisting of the extensions of 0 the form (4.25). This subset has dimension at least 5, since dim(H4 (X)) ≥ 4 and we have a 1-dimensional parameter given by the choice of x in C. `f s The closure of MX (2, 0, 4) and the closure of NX (2, 0, 4) in MX (2, 0, 4) meet along a subset containing the S-equivalence classes of sheaves ﬁtting into (4.13). References [AK80] Allen B. Altman and Steven L. Kleiman, Compactifying the Picard scheme, Adv. in Math. 35 (1980), no. 1, 50–112. [AO94] Vincenzo Ancona and Giorgio Ottaviani, Stability of special instanton bundles on P2n+1, Trans. Amer. Math. Soc. 341 (1994), no. 2, 677–693. ACM BUNDLES ON PRIME FANO THREEFOLDS 27

[AC00] Enrique Arrondo and Laura Costa, Vector bundles on Fano 3-folds without intermediate cohomology, Comm. Algebra 28 (2000), no. 8, 3899– 3911. [AF06] Enrique Arrondo and Daniele Faenzi, Vector bundles with no intermediate cohomology on Fano threefolds of type V22, Pacific J. Math. 225 (2006), no. 2, 201–220. [Arr07] Enrique Arrondo, A home-made Hartshorne-Serre correspondence, Rev. Mat. Complut. 20 (2007), no. 2, 423–443. [BF81] Constantin Banic˘ a˘ and Otto Forster, Sur les structures multiples, Manuscript, 1981. [Bea77] Arnaud Beauville, Variétésde Prym et jacobiennes intermédiaires, Ann. Sci. Ecole´ Norm. Sup. (4) 10 (1977), no. 3, 309–391. [Bea02] , Vector bundles on the cubic threefold, Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000), Contemp. Math., vol. 312, Amer. Math. Soc., Providence, RI, 2002, pp. 71–86. [BF07] Maria Chiara Brambilla and Daniele Faenzi, Vector bundles on Fano threefolds of genus 7 and Brill-Noether loci, Preprint available at the authors’ webpages, 2007. [BGS87] Ragnar-Olaf Buchweitz, Gert-Martin Greuel, and Frank-Olaf Schreyer, Cohen-Macaulay modules on hypersurface singularities. II, In- vent. Math. 88 (1987), no. 1, 165–182. [C˘al00] Andrei Cald˘ araru˘ , Derived Categories of Twisted Sheaves on Calabi-Yau Manifolds, Ph. D. thesis, http://www.math.wisc.edu/ andreic/, 2000. [C˘al02] , Nonfine moduli spaces of sheaves on K3 surfaces, Int. Math. Res. Not. (2002), no. 20, 1027–1056. [CH04] Marta Casanellas and Robin Hartshorne, Gorenstein biliaison and ACM sheaves, J. Algebra 278 (2004), no. 1, 314–341. [CF06] Luca Chiantini and Daniele Faenzi, Rank 2 arithmetically Cohen- Macaulay bundles on a general quintic surface, Math. Nachr., to appear, 2006. [CM00] Luca Chiantini and Carlo Madonna, ACM bundles on a general quintic threefold, Matematiche (Catania) 55 (2000), no. 2, 239–258 (2002), Dedi- cated to Silvio Greco on the occasion of his 60th birthday (Catania, 2001). [CM04] , A splitting criterion for rank 2 bundles on a general sextic threefold, Internat. J. Math. 15 (2004), no. 4, 341–359. [Dru00] Stephane´ Druel, Espace des modules des faisceaux de rang 2 semi-stables 4 de classes de Chern c1 = 0, c2 = 2 et c3 = 0 sur la cubique de P , Internat. Math. Res. Notices (2000), no. 19, 985–1004. [EH88] David Eisenbud and Jurgen¨ Herzog, The classification of homogeneous Cohen-Macaulay rings of finite representation type, Math. Ann. 280 (1988), no. 2, 347–352. [Fae07] Daniele Faenzi, Bundles over Fano threefolds of type V22, Ann. Mat. Pura Appl. (4) 186 (2007), no. 1, 1–24. [Fae08] , Rank 2 arithmetically Cohen-Macaulay bundles on a nonsingular cubic surface, J. Algebra 319 (2008), no. 1, 143–186. [GLN06] Laurent Gruson, Fatima Laytimi, and Donihakkalu S. Nagaraj, On prime Fano threefolds of genus 9, Internat. J. Math. 17 (2006), no. 3, 253– 261. [Gus82] N. P. Gushel0, Fano varieties of genus 6, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 6, 1159–1174, 1343. [Gus83] , Fano varieties of genus 8, Uspekhi Mat. Nauk 38 (1983), no. 1(229), 163–164. [Gus92] , Fano 3-folds of genus 8, Algebra i Analiz 4 (1992), no. 1, 120–134. [Har74] Robin Hartshorne, Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974), 1017–1032. [Har77] , Algebraic geometry, Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, No. 52. 28 MARIA CHIARA BRAMBILLA AND DANIELE FAENZI

[Har78] , Stable vector bundles of rank 2 on P3, Math. Ann. 238 (1978), no. 3, 229–280. [Har80] , Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121–176. [Hop84] Hans Jurgen¨ Hoppe, Generischer Spaltungstyp und zweite Chernklasse sta- biler Vektorraumbündelvom Rang 4 auf P4, Math. Z. 187 (1984), no. 3, 345–360. [Hor64] Geoffrey Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc. (3) 14 (1964), 689–713. [HL97] Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braun- schweig, 1997. [IM07a] Atanas Iliev and Laurent Manivel, Pfaffian lines and vector bundles on Fano threefolds of genus 8, J. Algebraic Geom. 16 (2007), no. 3, 499–530. [IM07b] , Prime Fano threefolds and integrable systems, Math. Ann. 339 (2007), no. 4, 937–955. [IM00a] Atanas Iliev and Dimitri Markushevich, The Abel-Jacobi map for a cubic threefold and periods of Fano threefolds of degree 14, Doc. Math. 5 (2000), 23–47 (electronic). [IM00b] , Quartic 3-fold: Pfaffians, vector bundles, and half- canonical curves, Michigan Math. J. 47 (2000), no. 2, 385–394, http://arXiv.org/abs/math.AG/0104203 [IM04] , Elliptic curves and rank-2 vector bundles on the prime Fano threefold of genus 7, Adv. Geom. 4 (2004), no. 3, 287–318. [IM07c] , Parametrization of sing Θ for a Fano 3-fold of genus 7 by moduli of vector bundles, Asian J. Math. 11 (2007), no. 3, 427–458. [IR05] Atanas Iliev and Kristian Ranestad, Geometry of the Lagrangian Grass- mannian LG(3, 6) with applications to Brill-Noether loci, Michigan Math. J. 53 (2005), no. 2, 383–417. [Isk77] Vasilii A. Iskovskih, Fano threefolds. I, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 3, 516–562, 717. [Isk78] , Fano threefolds. II, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 3, 506–549, English translation in Math. U.S.S.R. Izvestija 12 (1978) no. 3 , 469–506 (translated by Miles Reid). [IP99] Vasilii A. Iskovskikh and Yuri. G. Prokhorov, Fano varieties, Alge- braic geometry, V, Encyclopaedia Math. Sci., vol. 47, Springer, Berlin, 1999, pp. 1–247. [Knö87] Horst Knorrer¨ , Cohen-Macaulay modules on hypersurface singularities. I, Invent. Math. 88 (1987), no. 1, 153–164. [KO89] Siegmund Kosarew and Christian Okonek, Global moduli spaces and simple holomorphic bundles, Publ. Res. Inst. Math. Sci. 25 (1989), no. 1, 1–19. [Kuz96] Alexander G. Kuznetsov, An exceptional set of vector bundles on the varieties V22, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1996), no. 3, 41–44, 92. [Kuz05] , Derived categories of the Fano threefolds V12, Mat. Zametki 78 (2005), no. 4, 579–594, English translation in Math. Notes 78, no. 3-4, 537– 550 (2005). [LO87] Martin Lubke¨ and Christian Okonek, Moduli spaces of simple bundles and Hermitian-Einstein connections, Math. Ann. 276 (1987), no. 4, 663–674. [Mad00] Carlo Madonna, Rank-two vector bundles on general quartic hypersurfaces 4 in P , Rev. Mat. Complut. 13 (2000), no. 2, 287–301. [Mad02] , ACM vector bundles on prime Fano threefolds and complete intersection Calabi-Yau threefolds, Rev. Roumaine Math. Pures Appl. 47 (2002), no. 2, 211–222 (2003). [Man93] Yuri I. Manin, Notes on the arithmetic of Fano threefolds, Compositio Math. 85 (1993), no. 1, 37–55. ACM BUNDLES ON PRIME FANO THREEFOLDS 29

[Man86] Nicolae Manolache, Cohen-Macaulay nilpotent structures, Rev. Roumaine Math. Pures Appl. 31 (1986), no. 6, 563–575. [Man94] , Multiple structures on smooth support, Math. Nachr. 167 (1994), 157–202. [MT01] Dimitri Markushevich and Alexander S. Tikhomirov, The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold, J. Al- gebraic Geom. 10 (2001), no. 1, 37–62. [Mar80] Masaki Maruyama, Boundedness of semistable sheaves of small ranks, Nagoya Math. J. 78 (1980), 65–94. [MKRR07a] N. Mohan Kumar, Prabhakar A. Rao, and Girivau V. Ravindra, Arithmetically Cohen-Macaulay bundles on hypersurfaces, Comment. Math. Helv. 82 (2007), no. 4, 829–843. [MKRR07b] , Arithmetically Cohen-Macaulay bundles on three dimensional hypersurfaces, Int. Math. Res. Not. IMRN (2007), no. 8, Art. ID rnm025, 11. [Mo˘ı67] Boris G. Mo˘ıˇsezon, Algebraic homology classes on algebraic varieties, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 225–268. [MU83] Shigeru Mukai and Hiroshi Umemura, Minimal rational threefolds, Al- gebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 490–518. [Muk84] Shigeru Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math. 77 (1984), no. 1, 101–116. [Muk88] , Curves, K3 surfaces and Fano 3-folds of genus ≤ 10, Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 357– 377. [Muk89] , Biregular classiﬁcation of Fano 3-folds and Fano manifolds of coin- dex 3, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 9, 3000–3002. [Muk95] , Curves and symmetric spaces. I, Amer. J. Math. 117 (1995), no. 6, 1627–1644. [Ott89] Giorgio Ottaviani, Some extensions of Horrocks criterion to vector bundles on Grassmannians and quadrics, Ann. Mat. Pura Appl. (4) 155 (1989), 317– 341. [Pro90] Yuri G. Prokhorov, Exotic Fano varieties, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1990), no. 3, 34–37, 111. [Sch01] Frank-Olaf Schreyer, Geometry and algebra of prime Fano 3-folds of genus 12, Compositio Math. 127 (2001), no. 3, 297–319. E-mail address: [email protected], [email protected]

Dipartimento di Matematica “G. Castelnuovo”, Universita` di Roma Sapienza, Piazzale Aldo Moro 5, I-00185 Roma - Italia URL: http://web.math.unifi.it/users/brambill/

E-mail address: [email protected]

Universite´ de Pau et des Pays de l’Adour, Av. de l’universite´ - BP 576 - 64012 PAU Cedex - France URL: http://web.math.unifi.it/users/faenzi/